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\begin{abstract}

The objective of this study was to model the volume expansion factor ($VEF$), defined as being the ratio between total aboveground woody volume and stem merchantable volume, as a function of tree height and circumference at breast height. 

A large database (dataset \#1) of detailed stem and branches volume measurements, constituted by 8192 trees from 19 temperate tree species, was used for calibrating the models. In addition, an independent dataset (dataset \#2), constituted by 176 trees from 13 species, was collected for validating the models.

From dataset \#1, the RMSE for the prediction of the total volume varied from 0.009 to 0.469 $m^{3}$, depending on the circumference class, and the relative RMSE varied from 7.4\% to 13.7\%. 
A ten-fold cross-validation on dataset \#1 gave an average RMSE of 0.136 for the prediction of the $VEF$ and of 0.149 $m^{3}$ for the prediction of the total volume.  
Validation on dataset \#2 gave satisfactory results, even for the application of $VEF$ to other tree species. The largest errors were obtained when the model was clearly used in extrapolation, e.g., for five large-size \textit{Fraxinus} (whereas dataset \#1 included smaller trees) and several \textit{Quercus} and \textit{Fagus} from coppice-with-standards stands (whereas dataset \#1 included mainly pure even-aged high-forest trees).

The observed differences between species seemed consistent with the general knowledge about species-specific traits. For a same stem volume, angiosperms were found to have a much larger volume of branches than gymnosperms. For a tree of 100 cm in circumference, the lowest values of $VEF$ were obtained for \textit{Picea} and \textit{Abies} ($VEF < 1.1$) and the highest ones for \textit{Fagus}, \textit{Fraxinus} and \textit{Carpinus} ($VEF > 1.3$).

The methodology that was developed is easily applicable to other definitions of $VEF$ or biomass expansion factors ($BEF$). 
\end{abstract}


\begin{keyword}
Volume \sep Biomass \sep VEF \sep BEF \sep Non-linear mixed model

\end{keyword}
